Simplify; express your answer in exponential form. Assume $r\neq 0, z\neq 0$. $\dfrac{{(r^{4}z^{-2})^{4}}}{{(rz)^{-5}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(r^{4}z^{-2})^{4} = (r^{4})^{4}(z^{-2})^{4}}$ On the left, we have ${r^{4}}$ to the exponent ${4}$ . Now ${4 \times 4 = 16}$ , so ${(r^{4})^{4} = r^{16}}$ Apply the ideas above to simplify the equation. $\dfrac{{(r^{4}z^{-2})^{4}}}{{(rz)^{-5}}} = \dfrac{{r^{16}z^{-8}}}{{r^{-5}z^{-5}}}$ Break up the equation by variable and simplify. $\dfrac{{r^{16}z^{-8}}}{{r^{-5}z^{-5}}} = \dfrac{{r^{16}}}{{r^{-5}}} \cdot \dfrac{{z^{-8}}}{{z^{-5}}} = r^{{16} - {(-5)}} \cdot z^{{-8} - {(-5)}} = r^{21}z^{-3}$